Displacement of electron in magnetic and electric field, verification of one logic

I have solved this problem, but just need verification of my thinking about this as well as the question book has used full page for solution(using simple harmonic) but I had done it in a rather different and quicker way

Q– An electron is released from origin where a uniform electric field exists in negative y direction and a uniform magnetic field in z direction find its displacement from y axis line when velocity of electron is perpendicular to electric field for the first time (see the diagram blue lines are electric field ones
crosses in x-y plane represents magnetic field )

my logic that I have used to solve it(the answer comes out to be correct using this logic)—————– since work done by magnetic field on any charged particle inside it is $0$
so only work done is by electric field

by work energy theorem—-

$int eEdy$ $=$ $frac{mv^2}{2}$—-1 (no work done by magnetic force is considered) Note :- $e$ is charge of electron

it is from logic that the magnetic force is always perpendicular to charge velocity, hence its displacement so work done by it must be $0$ in this case as well isn’t it ? so the above equation –1 is correct or not??? I guess it correct as answer comes out to be same

if I am correct then it would save my brain as well as time from being wrecked by the full page mind boggling solution that the book has stated
and as I said earlier I had solved it , I just want to verify my logic , as the book took a huge difficult solution to answer it , any help would save me